Exotic Options

Options, Futures and Derivative Securities

Lorenzo Naranjo

Spring 2025

Packages

A package is a portfolio of standard options.
The main difference between a package and an option strategy is that the package is sold as a whole product, whereas an option strategy involves trading different options at the same time.
We have studied many option strategies such as bull spreads, bear spreads, straddles, strangles, butterflies, and condors, which could be sold as a package.

Range Forward Contracts

One popular package is a range forward contract.
Have the effect of ensuring that the exchange rate paid or received will lie within a certain range.
When currency is to be paid, it involves selling a put with strike K_{1} and buying a call with strike K_{2} (with K_{1} < K_{2}).
- This would be similar to a long forward position.
When currency is to be received it involves buying a put with strike K_{1} and selling a call with strike K_{2}.
- This would be similar to a short forward position.
Normally the price of the put equals the price of the call so the contract has zero cost.

Variations of the Black & Scholes Framework

Gap Options

A gap call pays S_{T} - \textcolor{blue}{K_{1}} when S_{T} > \textcolor{red}{K_{2}}, and zero otherwise.
A gap put pays \textcolor{blue}{K_{1}} - S_{T} when S_{T} < \textcolor{red}{K_{2}}, and zero otherwise.
We can adapt our previous analysis to get: \begin{align*} \text{Gap Call} & = S e^{-\delta T} \mathop{\Phi}(d_{1}) - \textcolor{blue}{K_{1}} e^{-r T} \mathop{\Phi}(d_{2}) \\ \text{Gap Put} & = \textcolor{blue}{K_{1}} e^{-r T} \mathop{\Phi}(-d_{2}) - S e^{-\delta T} \mathop{\Phi}(-d_{1}) \end{align*} where \begin{align*} d_{1} & = \frac{\ln(S/\textcolor{red}{K_{2}}) + (r - \delta + 0.5 \sigma^{2}) T}{\sigma \sqrt{T}} \\ d_{2} & = d_{1} - \sigma \sqrt{T} \end{align*}

Example 1

Suppose you want to price an option that pays S_{T} - 180 if S_{T} > 220 in 10 months.
The option is written on a non-dividend paying asset whose current price is $200 and that has a volatility of returns of 40%.
The risk-free rate is 6% per year with continuous compounding.

Example 1 (cont’d)

The option in the example is a gap call option where K_{1} = 180 and K_{2} = 220.
This implies, \begin{align*} d_{1} & = \frac{\ln(200/220) + (0.06 + 0.5 (0.40)^{2}) (10/12)}{0.40 \sqrt{10/12}} = 0.0585 \\ d_{2} & = d_{1} - 0.40 \sqrt{10/12} = -0.3067 \\ \end{align*}
The gap call price is then given by, \text{Gap Call} = 200 \times \mathop{\Phi}(0.0585) - 180 e^{-0.06 \times 10/12} \times \mathop{\Phi}(-0.3067) = \$39.68

Binary Options

A call pays Q if S_{T} > K, otherwise pays nothing. \text{Value} = Q e^{-r T} \mathop{\Phi}(d_{2})
A put pays Q if S_{T} < K, otherwise pays nothing. \text{Value} = Q e^{-r T} \mathop{\Phi}(-d_{2})
An call pays S_{T} if S_{T} > K, otherwise pays nothing. \text{Value} = S e^{-\delta T} \mathop{\Phi}(d_{1})
An put pays S_{T} if S_{T} < K, otherwise pays nothing. \text{Value} = S e^{-\delta T} \mathop{\Phi}(-d_{1})

Example 2

Consider a stock that pays a dividend yield of 3% and that has a volatility of returns of 35%.
The stock price is $150 and the risk-free rate is 8%.
First, let’s price an asset-or-nothing call that pays 1 share of the stock if the stock price in 6 months is above $150.
We can first compute d_{1} = \frac{\ln(150/150) + (0.08 - 0.03 + 0.5 \times 0.35^{2}) \times 0.5}{0.35 \sqrt{0.5}} = 0.2248
The costs of the asset-or-nothing call is 150 e^{-0.03 \times 0.5} \mathop{\Phi}(0.2248) = \$87.02.

Forward Start Options

The option starts at a future time \tau and expires at time T > \tau.
Implicit in employee stock option plans.
Often structured so that strike price equals the underlying asset price at time \tau, that is, K = S_{\tau}.
- We do not know the strike price today!
Therefore, the value of a call or put option at time \tau is: \begin{align*} C_{\tau} & = S_{\tau} e^{-\delta (T - \tau)} \mathop{\Phi}(d_{1}) - S_{\tau} e^{-r (T - \tau)} \mathop{\Phi}(d_{2}) \\ P_{\tau} & = S_{\tau} e^{-r (T - \tau)} \mathop{\Phi}(-d_{2}) - S_{\tau} e^{-\delta (T - \tau)} \mathop{\Phi}(-d_{1}) \end{align*} where d_{1} = \dfrac{(r - \delta + 0.5 \sigma^{2}) (T - \tau)}{\sigma \sqrt{T - \tau}} and d_{2} = d_{1} - \sigma \sqrt{T - \tau}.

Valuing a Forward Start Option

The price of the option today is just V = \operatorname{E}(V_{\tau}) e^{-r \tau} where the expectation is taken of course with respect the risk-neutral measure.
Noting that the futures price of a contract expiring at time \tau is given by F = \operatorname{E}(S_{\tau}) = S e^{(r - \delta) \tau}, we have that \operatorname{E}(S_{\tau}) e^{-r \tau} = S e^{-\delta \tau}.
Therefore: \begin{align*} C & = \left(S e^{-\delta (T - \tau)} \mathop{\Phi}(d_{1}) - S e^{-r (T - \tau)} \mathop{\Phi}(d_{2})\right) e^{-\delta \tau} \\ P & = \left(S e^{-r (T - \tau)} \mathop{\Phi}(-d_{2}) - S e^{-\delta (T - \tau)} \mathop{\Phi}(-d_{1})\right) e^{-\delta \tau} \end{align*} where d_{1} = \dfrac{(r - \delta + 0.5 \sigma^{2}) (T - \tau)}{\sigma \sqrt{T - \tau}} and d_{2} = d_{1} - \sigma \sqrt{T - \tau}.
We can then see that the value of a forward start option is e^{-\delta \tau} times the value of similar option starting today.

Cliquet Option

A series of call or put options with rules determining how the strike price is determined.
For example, a cliquet might consist of 20 at-the-money three-month options. The total life would then be five years.
When one option expires a new similar at-the-money is coming into existence.
As you can see, this would be a portfolio of 20 forward starting options that we just saw how to value.

Chooser Options

Option starts at time 0 and matures at T.
At time \tau (0 < \tau < T) the buyer chooses whether it is a put or call with strike K and expiring at T, at which point the value of the chooser is \max(C_{\tau}, P_{\tau}).
From put-call parity: P_{\tau} = C_{\tau} + K e^{-r (T - \tau)} - S_{\tau} e^{-\delta (T - \tau)} which implies that: \max(C_{\tau}, P_{\tau}) = C_{\tau} + e^{-\delta (T - \tau)} \max\left(K e^{-(r - \delta)(T - \tau)} - S_{\tau}, 0\right)
This is the payoff of a call with strike K and expiring at T plus e^{-q (T - \tau)} puts with strike \widetilde{K} = K e^{-(r - \delta)(T - \tau)} and expiring at time \tau.

Compound Option

Option to buy or sell an option.
We have therefore four possible combinations:
- Call on call
- Put on call
- Call on put
- Put on put
These options can be valued analytically (we will not cover this in class, though).
Intuitively, the price of such options is quite low compared with the underlying option.

Path-Dependent Options

Lookback Options

A floating lookback call pays S_{T} - S_{\textit{min}} at time T.
- Allows the buyer to buy the stock at the lowest observed price in some interval of time.
A floating lookback put pays S_{\textit{max}} - S_{T} at time T.
- Allows the buyer to sell the stock at the highest observed price in some interval of time.
A fixed lookback call pays \max(S_{\textit{max}} - K, 0) at time T.
- Like a regular call but the final payoff depends on the maximum value of the stock during the lifetime of the option.
A fixed lookback put pays \max(K - S_{\textit{min}}, 0) at time T.
- Like a regular put bu the final payoff depends on the minimum value during the life of the option.
It is possible to derive analytic formulas for all types.

Example 3

Consider a non-dividend paying stock that trades for $100.
Every 3-months, the stock price can increase or decrease by 5%.
The risk-free rate is 6% per year with continuous compounding.
We will compute the price of a floating lookback put that pays S_{\textit{max}} - S_{T} in 6 months.

Example 3 (cont’d)

For lookback options is better to draw the full tree for the stock in order to see all four possible histories for the stock price.
We can then compute the maximum stock price of each history and then the final payoff of the option.

Example 3 (cont’d)

The risk-neutral probability of an up-move is q = \frac{e^{-0.06 \times 3/12} - 0.95}{1.05 - 0.95} = 0.6511
The tree for the lookbak put is presented below.

Example 3 (cont’d)

We can then compute: \begin{align*} L_{u} & = (0 q + 5.25 (1 - q)) e^{-0.06 \times 3/12} = 1.80 \\ L_{d} & = (0.25 q + 9.75 (1 - q)) e^{-0.06 \times 3/12} = 3.51 \\ L & = (1.80 q + 3.51 (1 - q)) e^{-0.06 \times 3/12} = 2.36 \end{align*}
Alternatively, we could obtain the price of the lookback put directly from the final payoffs: L = (0 q^{2} + 5.25 q (1 - q) + 0.25 (1 - q) q + 9.75 (1 - q)^{2}) e^{-0.06 \times 6/12} = 2.36

Asian Options

The payoff of such options is related to the average stock price \bar{S} from time 0 until T.
Average price options pay:
- Call: \max(\bar{S} - K, 0)
- Put: \max(K - \bar{S}, 0)
Average strike options pay:
- Call: \max(S_{T} - \bar{S}, 0)
- Put: \max(\bar{S} - S_{T}, 0)
No exact analytic valuation, but can be approximately valued by assuming that the average stock price is lognormally distributed.

Example 4

Consider a non-dividend paying stock that trades for $100.
Every 3-months, the stock price can increase or decrease by 5%.
The risk-free rate is 6% per year with continuous compounding.
We will compute the price of an average price call that pays \bar{S}_{T} - K in 6 months, where K = 100.

Example 4 (cont’d)

As we did for lookback options, it is better to draw the full tree for the stock in order to see all four possible histories for the stock price.
We can then compute the average stock price of each history and then the final payoff of the option.

Example 4 (cont’d)

The risk-neutral probability of an up-move is q = \frac{e^{-0.06 \times 3/12} - 0.95}{1.05 - 0.95} = 0.6511
The tree for the Asian call is presented below.

Example 4 (cont’d)

We can then compute: \begin{align*} A_{u} & = (5.08 q + 1.58 (1 - q)) e^{-0.06 \times 3/12} = 3.80 \\ A_{d} & = (0 q + 0 (1 - q)) e^{-0.06 \times 3/12} = 0 \\ A & = (3.80 q + 0 (1 - q)) e^{-0.06 \times 3/12} = 2.44 \\ \end{align*}
Alternatively, we could obtain the price of the lookback put directly from the final payoffs: L = (5.08 q^{2} + 1.58 q (1 - q) + 0 (1 - q) q + 0 (1 - q)^{2}) e^{-0.06 \times 6/12} = 2.44

Barrier Options

Barrier options are either call or put options that get activated or deactivated depending on whether the stock hits a barrier from above or below.
- “In” options come into existence only if stock price hits the barrier before option maturity.
- “Out” options die if stock price hits the barrier before option maturity.
- “Up” options require that the stock hits the barrier from below.
- “Down” options require that the stock hit the barrier form above.
Therefore, there are eight possible combinations.

Shout Options

Buyer can shout once during the life of the option.
For a call option, the final payoff is the maximum between:
- Usual option payoff, \max(S_{T} - K, 0), or
- Intrinsic value at time of shout \tau, S_{\tau} - K.
Payoff: \max(\max(S_{T} - K, 0), S_{\tau} - K) = \max(S_{T}- S_{\tau}, 0) + S_{\tau} - K
Similar to lookback option but cheaper.

Other Exotic Options

Exchange Options

Option to exchange one asset for another.
For example, an option to exchange one unit of U for one unit of V.
Payoff is then \max(V_{T} - U_{T}, 0).

Basket Options

A basket option is an option to buy or sell a portfolio of assets.
This can be valued by calculating the first two moments of the value of the basket at option maturity and then assuming it is lognormal.

Non-Standard American Options

Exercisable only on specific dates (Bermudans)
Early exercise allowed during only part of life (initial “lock out” period)
Strike price changes over the life (warrants, convertibles)